The aim of the general approximation methods concerning linear positive operators is to deal with convergence behavior. The accuracy can be ascertained to a desired degree by applying different methods. We are also concerned with the amount of computation required to achieve this accuracy. A direct theorem provides the order of approximation for functions of specified smoothness. The converse of the direct result, that is, the inverse theorem, infers the nature of smoothness of the function from its order of approximation. Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations that arise in the mathematical modeling of real-world phenomena. The rate of convergence infers the speed at which a convergent sequence approaches its limit.
This work treats the convergence results mainly for linear positive operators. After the well-known theorem due to Weierstrass and the important convergence theorem of Korovkin, many new operators were proposed and constructed by several researchers. The theory of these operators has been an important area of research in the last few decades. The basic results and direct estimates in both local and global approximation are also presented. We also discuss the asymptotic expansion of some of the linear positive operators, which is important for convergence estimates. We know that to improve the order of approximation, we can consider the combinations. By considering the linear combinations,we have to slacken the positivity conditions of the operators. Here we discuss linear and iterative combinations and present some results. Some operators reproduce constant and linear functions. Two decades ago it was observed that if we modify the original operators, we can have a better approximation. Later some researchers studied this direction and observed that some operators that do not even preserve linear functions can give a better approximation if they are modified to preserve linear functions. The overconvergence phenomena for certain operators, which were not discussed in the book by Gal, [77] are also discussed here. We present some results of overconvergencies to larger sets in the complex plane.
In this book, the crucial role of the rate of convergence for functions of bounded variation and for functions having derivatives of bounded variation is emphasized. New and efficient methods that are applicable to general operators are also discussed. The advantages of these methods consist of obtaining improved and even optimal estimates, as well as of broadening the applicability of the results. Several results have been established for different exponential-type operators and inegral operators. Still, this type of study can be extended to other generalizations of the known operators. Also, the rate of convergence for functions of bounded variation on mixed summation-integral-type operators having different basis functions has not been studied to date, including Szász-Baskakov, Baskakov-Szász, and Beta-Szász operators, among others. It can be considered an open problem. We mention some results without proofs, while in other cases, proofs and outlines are given. The book is useful for beginners and for those who are working on analysis and related areas.
This monograph is divided into 11 chapters:
In Chap. 1, which is introductory in nature, we present some basic definitions and the standard theorems, which are important for the convergence point of view.
In Chap. 2, we discuss some important results on linear positive operators related to convergence. Some direct results, which include pointwise convergence, asymptotic formulas, and error estimations, are presented here. We also deal with some important results, including discretely defined operators, Kantorovich- and Durrmeyer-type operators, mixed summation-integral-type operators, Phillips operators, and other integral-type operators.
In Chap. 3, we discuss the asymptotic behavior of some of the linear positive operators. We discuss the complete asymptotic expansion of the Baskakov-Kantorovich, Szász-Baskakov, Meyer-König-Zeller-Durrmeyer, and Beta operators.
In Chap. 4, we mention approximation for certain combinations. The linear combinations are no longer positive operators. Here we study some results for linear and iterative combinations. Also, we consider a different form of the linear combinations and present direct estimates for combinations of Szász-Baskakov operators.
In Chap. 5, we present the techniques for getting a better approximation. Many well-known approximating operators Ln, preserve the constant as well as linear functions. In 1983, King considered the modification of the classical Bernstein polynomials so that the modified form preserves the test function e2. A better approximation can be achieved for the modified form. In this chapter, we present some results for different operators discussed in recent years.
In Chap. 6, we study the overconvergence phenomenon for certain operators that were not discussed in the book by Gal [77]. We present some results of overconvergencies to larger sets in the complex plane than in the real domain. We discuss some very recent results on complex Baskakov-Stancu operators, complex Favard-Szász-Mirakjan-Stancu operators, complex Beta operators of the second kind, genuine Durrmeyer-Stancu polynomials, and certain complex Durrmeyertype operators.
In Chap. 7, the pointwise approximation properties of some approximation operators of the probabilistic type are established and studied. The rates of convergence for Legendre-Fourier series, Hermite-Fejér polynomials, exponentialtype operators (which include Bernstein, Szász, Baskakov, Gamma, and Weierstrass operators) are also discussed. The rates of convergence of such operators for a bounded variation function f are given at those points x, where f(x+) and f(x-) exist. Here we also present some results on the Durrmeyer variants of such operators. We also present the related results for a general class of summation-integral-type operators and the Meyer-König and Zeller operators.
In Chap. 8, we discuss various Bézier variants of the approximation operators. They have close relationships with the fields of geometry modeling and design. In this chapter, we present some important results on the approximation of Bézier variants of a series of approximation operators, which include Bézier variants of the Bernstein, Bleimann-Butzer-Hann, Balazs-Kantorovich, Szász-Kantorovich, Baskakov, Baskakov-Kantorovich, Baskakov-Durrmeyer, and MKZ operators, among others.
In Chap. 9, we study some other related results. We present the rate of approximation on nonlinear operators. Also, we discuss the rate of convergence in terms of Chanturiya's modulus of variation. We also present some results for bounded and absolutely continuous functions. Finally, we present the rate of convergence for functions having derivatives coinciding a.e. with the function of bounded variation.
In Chap.10, we present results related to simultaneous approximation. Although lot of papers have appeared on such topic, in this chapter we mention the rate of simultaneous approximation for certain Durrmeyer-Bézier operators for functions of bounded variation. We also present the rate of convergence in simultaneous approximation for certain summation-integral-type operators having derivatives of bounded variation.
In the brief final chapter, Chap. 11, we present the future scope of related topics and mention some open problems.

1 Preliminaries 1

1.1 Korovkin's Theorem 2

1.2 Weierstrass Approximation Theorems 4

1.3 Order of Approximation 8

1.4 Differential Properties of Function 11

1.5 Notations and Inequalities 13

1.6 Bounded Variation 15

2 Approximation by Certain Operators 17

2.1 Discretely Defined Operators 17

2.2 Kantorovich Operators 27

2.3 Durrmeyer-Type Operators 29

2.4 Szász-Beta-Type Operators 36

2.5 Phillips Operators 52

2.6 Integral Modification of Jain Operators 60

2.7 Generalized Bernstein-Durrmeyer Operators 64

2.8 Generalizations of Baskakov Operators 72

2.9 Mixed Summation-Integral Operators 89

3 Complete Asymptotic Expansion 93

3.1 Baskakov-Kantorovich Operators 93

3.2 Baskakov-Szász-Durrmeyer Operators 97

3.3 Meyer-König-Zeller-Durrmeyer Operators 100

3.4 Beta Operators of the First Kind 103

4 Linear and Iterative Combinations 109

4.1 Linear Combinations 109

4.2 Iterative Combinations 118

4.3 Another Form of Linear Combinations 129

4.4 Combinations of Szász-Baskakov Operators 132

5 Better Approximation 141

5.1 Bernstein-Durrmeyer-Type Operators 142

5.2 Phillips Operators 146

5.3 Szász-Mirakjan-Beta Operators 147

5.4 Integrated Szász-Mirakjan Operators 149

5.5 Beta Operators of the Second Kind 151

6 Complex Operators in Compact Disks 155

6.1 Complex Baskakov-Stancu Operators 155

6.2 Complex Favard-Szász-Mirakjan-Stancu Operators 166

6.3 Complex Beta Operators of the Second Kind 175

6.4 Genuine Durrmeyer-Stancu Polynomials 189

6.5 New Complex Durrmeyer Operators 196

6.6 Complex q-Durrmeyer-Type Operators 206

6.7 Complex q-Bernstein-Schurer Operators 210

7 Rate of Convergence for Functions of Bounded Variation 213

7.1 Fourier and Fourier-Legendre Series 213

7.2 Hermite-Fejér Polynomials 216

7.3 Exponential-Type Operators 217

7.4 Bernstein-Durrmeyer-Type Polynomials 222

7.5 Szász-Mirakyan-Durrmeyer-Type Operators 226

7.6 Baskakov-Durrmeyer-Type Operators 231

7.7 Baskakov-Beta Operators 239

7.8 General Summation-Integral-Type Operators 242

7.9 Meyer-König-Zeller Operators 244

8 Convergence for Bounded Functions on Bézier Variants 249

8.1 Bernstein-Bézier-Type Operators 250

8.2 Bleimann-Butzer-Hann-Bézier Operators 252

8.3 Balazs-Kantorovich-Bézier Operators 254

8.4 Szász-Kantorovich-Bézier Operators 257

8.5 Baskakov-Bézier Operators 264

8.6 Baskakov-Kantorovich-Bézier Operators 268

8.7 Baskakov-Durrmeyer-Bézier Operators 275

8.8 MKZ Bézier-Type Operators 280

9 Some More Results on the Rate of Convergence 287

9.1 Nonlinear Operators 287

9.2 Chanturiya's Modulus of Variation 296

9.3 Functions with Derivatives of Bounded Variation 301

9.4 Convergence for Bounded and Absolutely Continuous Functions 308

10 Rate of Convergence in Simultaneous Approximation 313

10.1 Bernstein-Durrmeyer-Bézier-Type Operators 314

10.2 General Class of Operators for DBV 325

10.3 Baskakov-Beta Operators for DBV 333

10.4 Szász-Mirakian-Stancu-Durrmeyer Operators 334

11 Future Scope and Open Problems 345

References 349

Index 359

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